Falling Bodies

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Definition

Also known as projectile motion. It is a formula that helps to find the vertical motion of an object that is throwing straight up or down, or simply dropped under gravitational force on Earth. The formula contains the variables of height, time, acceleration (due to gravity), initial veolocity and initial height.

Worked examples

\(h(t) = -16t^2 + v_0 t + h_0\)

Standard falling-body formula (feet): height after t seconds, starting at height \(h_0\) with initial velocity \(v_0\).

\(h(t) = -4.9t^2 + 20t + 5\)

Object thrown upward at 20 m/s from 5 m high; \(-4.9\) is half of gravity in meters per second squared.

\(h(2) = -16(2)^2 + 32(2) + 10 = -64 + 64 + 10 = 10\) ft\(\)

Plug in \(t=2\) to find the height at 2 seconds.

Common mistakes

\(h(t) = -16t^2 + v_0 t + h_0\) with \(v_0 = -10\) for dropping → \(v_0 = 0\) when simply dropped (no initial velocity) Dropped means released from rest; negative \(v_0\) would mean thrown downward.
Using \(-16\) when the problem gives meters → Use \(-4.9\) for meters, \(-16\) for feet The gravity constant depends on the unit system; always match the problem's units.
\(h(t) = -16t^2 + 10\) for object thrown up at 10 ft/s → \(h(t) = -16t^2 + 10t + h_0\) Initial velocity must multiply \(t\); the constant term is initial height, not velocity.

Where you'll use it next

Falling-body formulas reappear in physics for kinematics and energy problems, calculus when modeling motion and derivatives, and real applications like optimizing projectile launch angles or designing safety systems.

Found in 1 StudyPug lesson

Position velocity acceleration: Derivative

Calculus 1

We now know that taking the derivative of a function will give us the slope, or the instantaneous rate of change of the function. So what if we take the derivative of a function that models the position of some object moving along a line? It gives us its velocity! And if we differentiate its velocity function? It gives us its acceleration! In this section, we will study the relationship between position, velocity and acceleration using our knowledge of differential calculus.

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